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ANALYTICAL Introduction and Basic Concepts

Paweł FRITZKOWSKI, Ph.D. Eng.

Division of Technical Mechanics Institute of Faculty of Mechanical and Management POZNAN UNIVERSITY OF TECHNOLOGY Agenda

1 Introduction to the Course 2 Degrees of Freedom and Constraints 3 Generalized Quantities 4 Problems 5 Summary 6 Bibliography

Paweł Fritzkowski Introduction: Basic Concepts 2 / 30 Introduction to the Course – What is it all about?

Analytical mechanics... is a branch of results from a reformulation of the classical Galileo’s and Newton’s concepts is an approach different from the vector Newtonian mechanics: more advanced, sophisticated and mathematically-oriented eliminates the need to analyze on isolated parts of mechanical systems is a more global way of thinking: allows one to treat a system as a whole

Paweł Fritzkowski Introduction: Basic Concepts 3 / 30 Introduction to the Course Analytical mechanics – What is it all about? (cont.)

provides more powerful and easier ways to derive equations of , even for complex mechanical systems is based on some scalar functions which describe an entire system is a common tool for creating mathematical models for numerical simulations has spread far beyond the pure mechanics and influenced various areas of

Paweł Fritzkowski Introduction: Basic Concepts 4 / 30 Introduction to the Course Analytical mechanics – What is it all about?

Paweł Fritzkowski Introduction: Basic Concepts 5 / 30 Introduction to the Course Analytical mechanics – The key contributors

Joseph Louis Lagrange (1736-1813), mathematician and astronomer, born in Italy, worked mainly in France

Sir (1805-1865), Irish mathematician, and astronomer

Paweł Fritzkowski Introduction: Basic Concepts 6 / 30 Introduction to the Course Analytical mechanics – The key contributors

There already exist several treatises on mechanics, but the purpose of this one is entirely new. I propose to condense the theory of this science and the method of solving the related problems to general formulas whose simple application produces all the necessary equations for the solution of each problem. I hope that my presentation achieves this purpose and leaves nothing lacking. J.L. Lagrange, 1788

Paweł Fritzkowski Introduction: Basic Concepts 7 / 30 Introduction to the Course Analytical mechanics – The key contributors

The theoretical development of the laws of motion of bodies is a problem of such interest and importance, that it has engaged the attention of all the most eminent mathematicians, since the invention of as a mathematical science by Galileo, and especially since the wonderful extension which was given to that science by Newton. Among the successors of those illustrious men, Lagrange has perhaps done more than any other analyst, to give extent and harmony to such deductive researches, by showing that the most varied consequences respecting the of systems of bodies may be derived from one radical formula; the beauty of the method so suiting the dignity of the results, as to make of his great a kind of scientific poem. W.R. Hamilton, 1834

Paweł Fritzkowski Introduction: Basic Concepts 8 / 30 Introduction to the Course Lecture contents

1 Introduction and Basic Concepts 2 Static Equilibrium 3 Lagrangian Dynamics 4 Linear Oscillators 5 Nonlinear Dynamics and Chaos

Paweł Fritzkowski Introduction: Basic Concepts 9 / 30 Introduction to the Course Objectives of the course

To enrich your knowledge on mechanics with some elements of analytical mechanics, theory for discrete systems, nonlinear dynamics and chaos theory To shape your skills in mathematical modelling and analytical description of equilibrium and motion of complex mechanical systems To develop your ability to analyze motion of mechanical systems

Paweł Fritzkowski Introduction: Basic Concepts 10 / 30 Introduction to the Course Learning objectives

At the end of this course you should be able to... decide which method to use in order to mathematically describe motion of a given system determine equilibrium positions of mechanical systems derive for systems of rigid bodies analyze behaviour of mechanical systems and determine the character of motion

Paweł Fritzkowski Introduction: Basic Concepts 11 / 30 Introduction to the Course Related literature

1 Thornton S.T., Marion J.B., Classical Dynamics of Particles and Systems. Brooks/Cole, 2004. 2 Hand L.N., Finch J.D., Analytical Mechanics. Cambridge University Press, 1998. 3 Goldstein H., Poole Ch., Safko J., Classical Mechanics. Addison-Wesley, 2001. 4 Whittaker E.T., A Treatise on the of Particles and Rigid Bodies. Cambridge University Press, 1917. 5 Hilborn R.C., Chaos and Nonlinear Dynamics. An Introduction for Scientists and . Oxford University Press, 2000.

Paweł Fritzkowski Introduction: Basic Concepts 12 / 30 Degrees of Freedom and Constraints Mechanical system and its configuration

The radius vector of the body:

ri = ri(t) ,

where i = 1, 2, . . . , nb In three dimensions:

xi = xi(t)

yi = yi(t)

zi = zi(t)

Ci – the center of of the ith body nb – the number of members of the system

Paweł Fritzkowski Introduction: Basic Concepts 13 / 30 Degrees of Freedom and Constraints Degrees of freedom

Degrees of freedom (DOF) of a body – admissible elementary movements (translations or rotations) of the body Degrees of freedom of a system – admissible elementary movements of all members within the system Number of degrees of freedom s is equal to the minimal number of variables that can completely describe the configuration of the system

Paweł Fritzkowski Introduction: Basic Concepts 14 / 30 Degrees of Freedom and Constraints Constraints

Constraints – conditions that limit the motion of the system and reduce the number of degrees of freedom Constraints result from contact of the system with the environment and interconnections between particular members of the system Each source of constraints (supports, connectors, etc.) has its valence w, i.e. the number of constraints imposed on the system

Paweł Fritzkowski Introduction: Basic Concepts 15 / 30 Degrees of Freedom and Constraints Valence of typical supports in plane problems

Freely guide: w = 1 Rod, cable, rope: w = 1

Pin connection, joint: w = 2 Simple support: w = 2

Paweł Fritzkowski Introduction: Basic Concepts 16 / 30 Degrees of Freedom and Constraints Number of degrees of freedom

Assume that a mechanical system

– is composed of nb parts: np particles and nr rigid bodies (nb = np + nr) – is subjected to nc constraint sources with valences wk (k = 1, 2, . . . , nc) The number of degrees of freedom, s, can be calculated as follows: two-dimensional (plane) problems:

nc s = 2np + 3nr − ∑ wk k=1 three-dimensional (spatial) problems:

nc s = 3np + 6nr − ∑ wk k=1

Paweł Fritzkowski Introduction: Basic Concepts 17 / 30 Degrees of Freedom and Constraints Constraints

Mathematically constraints can be expressed as equations connecting the coordinates and/or the of the bodies:

gj(t, r1, r2,..., rnb ) = 0 , j = 1, 2, . . . , nc

or

gj(t, r1, r2,..., rnb , r˙1, r˙2,..., r˙nb ) = 0 , j = 1, 2, . . . , nc ,

where nc – the number of constraints

Paweł Fritzkowski Introduction: Basic Concepts 18 / 30 Degrees of Freedom and Constraints

Classification of constraints [Thornton & Marion, 2004]

Geometric or kinematic: 1 geometric constraints – the equations connect only the coordinates:

gj(t, r1, r2,..., rnb ) = 0

2 kinematic constraints – the equations connect both the coordinates and velocities:

gj(t, r1, r2,..., rnb , r˙1, r˙2,..., r˙nb ) = 0

Paweł Fritzkowski Introduction: Basic Concepts 19 / 30 Degrees of Freedom and Constraints

Classification of constraints [Thornton & Marion, 2004] (cont.)

Bilateral or unilateral: 1 bilateral constraints – expressed by equations:

gj(r1, r2,...) = 0

2 unilateral constraints – expressed by inequalities:

gj(t, r1, r2,...) > 0 or gj(t, r1, r2,...) ≥ 0

Paweł Fritzkowski Introduction: Basic Concepts 20 / 30 Degrees of Freedom and Constraints

Classification of constraints [Thornton & Marion, 2004] (cont.)

Scleronomic (fixed) or rheonomic (moving): 1 scleronomic constraints – the equations do not contain explicitly: gj(r1, r2,...) = 0 2 rheonomic constraints – the equations involve time explicitly:

gj(t, r1, r2,...) = 0

Paweł Fritzkowski Introduction: Basic Concepts 21 / 30 Generalized Quantities

Generalized coordinates [Hand & Finch, 1998]

Generalized coordinates – a set of variables

q1, q2,..., qn

that completely specify the configuration of a system If n = s, the generalized coordinates are independent (the so called proper set of generalized coordinates) If n > s, then only s generalized coordinates are independent (q1, q2,..., qs); the rest of them (qs+1, qs+2,..., qn) are dependent on the former ones

Paweł Fritzkowski Introduction: Basic Concepts 22 / 30 Generalized Quantities

Generalized coordinates [Hand & Finch, 1998] (cont.)

Generalized coordinates qj can be of a different nature: they can describe translational or rotational motion Usually it is possible to formulate the transformation equations:

ri = ri(t, q1,..., qn)

that means

xi = xi(t, q1,..., qn)

yi = yi(t, q1,..., qn)

zi = zi(t, q1,..., qn)

Paweł Fritzkowski Introduction: Basic Concepts 23 / 30 Generalized Quantities Generalized velocities

Similarly, a set of generalized velocities may be defined, i.e. the time derivatives of the coordinates:

q˙ 1, q˙ 2,..., q˙ n

where n ≥ s Transformation equations for velocities take the form:

r˙i = r˙i(t, q1,..., qn , q˙ 1,..., q˙ n)

that means

x˙ i = x˙ i(t, q1,..., qn, q˙ 1,..., q˙ n)

y˙ i = y˙ i(t, q1,..., qn, q˙ 1,..., q˙ n)

z˙ i = z˙ i(t, q1,..., qn, q˙ 1,..., q˙ n)

Paweł Fritzkowski Introduction: Basic Concepts 24 / 30 Generalized Quantities Generalized forces

Assume that a set of external forces F1, F2,..., Fnb acts on a mechanical system, i.e. the

Fi = [Fxi, Fyi, Fzi]

is applied to the ith member

One can define a generalized force Qj associated with qj as:

nb nb   ∂ri ∂xi ∂yi ∂zi Qj = ∑ Fi = ∑ Fxi + Fyi + Fzi i=1 ∂qj i=1 ∂qj ∂qj ∂qj

the nature of the quantity Qj is strictly related to the character of qj: " – force", "angle – "

Paweł Fritzkowski Introduction: Basic Concepts 25 / 30 Generalized Quantities Generalized quantities

Quantity Symbol Translation Rotation

Coordinate qj x [m] ϕ [rad]

Velocity q˙ j v = x˙ [m/s] ω = ϕ˙ [rad/s] 2 2 q¨j a = x¨ [m/s ] ε = ϕ¨ [rad/s ]

Force Qj F [N] M [Nm] 2 pj p [kg m/s] k [kg m /s]

Paweł Fritzkowski Introduction: Basic Concepts 26 / 30 Problems Determine the number of degrees of freedom of the given systems:

Paweł Fritzkowski Introduction: Basic Concepts 27 / 30 Problems

Derive expressions for the generalized forces acting on the given systems:

Paweł Fritzkowski Introduction: Basic Concepts 28 / 30 Summary Conclusions and final remarks

Analytical mechanics... can be regarded as a non-vector formulation of mechanics involves scalar generalized quantities of different nature which are treated equally, based on the same rules eliminates the need to take into account the internal forces (constraints forces)

Paweł Fritzkowski Introduction: Basic Concepts 29 / 30 Bibliography

Hand L.N., Finch J.D., Analytical Mechanics. Cambridge University Press, 1998. Thornton S.T., Marion J.B., Classical Dynamics of Particles and Systems. Brooks/Cole, 2004.

Paweł Fritzkowski Introduction: Basic Concepts 30 / 30